Generating a large, non-compressible data stream

ABSTRACT

Generating non-compressible data streams is disclosed, including: receiving an initialization parameter; determining a constrained prime number, wherein the constrained prime number comprises a plurality of component values, wherein each of the plurality of component values comprises a prime number, wherein each of the plurality of component values is different; and generating a non-compressible sequence based at least in part on the initialization parameter and the constrained prime number.

BACKGROUND OF THE INVENTION

Certain applications require various sets of data for testing purposes. While real user data can be used for testing, such data changes slowly and infrequently. As such, non-user data can be generated and used for testing. Conventionally, testing data is generated by hashing and/or cryptography techniques. However, generating testing data by hashing and/or cryptography techniques may be slow and inefficient.

Furthermore, in conventional systems, a master copy of a data stream is needed to verify another copy of the data stream. The master copy of the data stream can be compared to the other copy of the data stream to determine whether the values of the data stream to be verified match those of the master copy. However, it may not be feasible and/or too costly to maintain a master copy of each data stream that is to be verified.

In some conventional systems, data is automatically compressed before it is sent across a network to potentially reduce the amount of data to be sent over the network. However, it may not be desirable to compress data in certain testing environments in which it is desired to maintain the original size of the data.

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments of the invention are disclosed in the following detailed description and the accompanying drawings.

FIG. 1 is a diagram showing an embodiment of a system for performing quality assurance on a storage duplication server.

FIG. 2 is a diagram showing an embodiment of a quality assurance server.

FIG. 3 is a flow diagram showing an embodiment of a process of generating a sequence using an initialization parameter and a prime number.

FIG. 4 is a flow diagram showing an embodiment of a process of generating a data stream using prime numbers.

FIG. 5 is a flow diagram showing an embodiment of a process of verifying a data stream.

FIG. 6 is a diagram showing a table that includes sample constrained and unconstrained 32-bit primeM values to help explain how a primeM value is determined to be constrained or unconstrained.

FIG. 7 is a flow diagram showing an embodiment of a process for identifying a set of constrained prime numbers.

FIG. 8 is a flow diagram showing an embodiment of a process for generating a non-compressible sequence using an initialized parameter and a constrained prime number.

FIG. 9A is a diagram showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number.

FIG. 9B is a diagram showing a frequency analysis table for all component byte values of a sampling of 1,032 bytes of the sequence of FIG. 9A

FIG. 10A is a diagram showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number.

FIG. 10B is a diagram showing a frequency analysis table for all component byte values of a sampling of 1,032 bytes of the sequence of FIG. 10A.

FIG. 10C is a diagram showing a frequency analysis table for all component byte values of a sampling of 12,288 bytes of the sequence of FIG. 10A.

FIG. 11 is a diagram showing a table of accumulator (generated data) internal byte value rotations.

FIG. 12 is a flow diagram showing an embodiment of a process for generating a non-compressible data stream using two constrained prime numbers.

FIGS. 13A to 13D are diagrams showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number.

FIG. 13E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 13A to 13D.

FIGS. 14A to 14D are diagrams showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number.

FIG. 14E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 14A to 14D.

FIGS. 15A to 15G are diagrams showing a table that includes 32-bit values of a non-compressible data stream generated from merging two non-compressible sequences.

FIG. 15H is a diagram showing a frequency analysis table for all component byte values of 8,124 bytes of the non-compressible data stream of FIGS. 15A to 15G.

FIGS. 16A to 16D are diagrams showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number.

FIG. 16E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 16A to 16D.

FIGS. 17A to 17D are diagrams showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number.

FIG. 17E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 17A to 17D.

FIGS. 18A to 18G are diagrams showing a table that includes 32-bit values of a compressible data stream generated from merging two compressible sequences.

FIG. 18H is a diagram showing a frequency analysis table for all component byte values of 8,184 bytes of the compressible data stream of FIGS. 18A to 18G.

DETAILED DESCRIPTION

The invention can be implemented in numerous ways, including as a process; an apparatus; a system; a composition of matter; a computer program product embodied on a computer readable storage medium; and/or a processor, such as a processor configured to execute instructions stored on and/or provided by a memory coupled to the processor. In this specification, these implementations, or any other form that the invention may take, may be referred to as techniques. In general, the order of the steps of disclosed processes may be altered within the scope of the invention. Unless stated otherwise, a component such as a processor or a memory described as being configured to perform a task may be implemented as a general component that is temporarily configured to perform the task at a given time or a specific component that is manufactured to perform the task. As used herein, the term ‘processor’ refers to one or more devices, circuits, and/or processing cores configured to process data, such as computer program instructions.

A detailed description of one or more embodiments of the invention is provided below along with accompanying figures that illustrate the principles of the invention. The invention is described in connection with such embodiments, but the invention is not limited to any embodiment. The scope of the invention is limited only by the claims and the invention encompasses numerous alternatives, modifications and equivalents. Numerous specific details are set forth in the following description in order to provide a thorough understanding of the invention. These details are provided for the purpose of example and the invention may be practiced according to the claims without some or all of these specific details. For the purpose of clarity, technical material that is known in the technical fields related to the invention has not been described in detail so that the invention is not unnecessarily obscured.

Embodiments of data stream generation using prime numbers are described herein. An initialization parameter is received. In some embodiments, the initialization parameter is any value selected from a constrained address space. In various embodiments, a “sequence” refers to a sequence of values that is generated using an initialization parameter and a prime number. In some embodiments, a first sequence is generated using a first prime number and the initialization parameter. In some embodiments, a second sequence is generated using a second prime number and the initialization parameter. In some embodiments, the first prime number and the second prime number are selected based on a revision parameter that is received. In some embodiments, each of the first prime number and the second prime number is selected from a constrained modified set of prime numbers. A data stream is generated by merging (e.g., interleaving) the first sequence and the second sequence. In various embodiments, a “data stream” refers to a sequence of values that is determined by the merging (e.g., interleaving) of at least two sequences. In some embodiments, a data stream can be referred to as a “merged sequence.” In some embodiments, a data stream is not deduplicatable. In various embodiments, a non-deduplicatable data stream comprises a data stream that does not include duplicate blocks of data (e.g., that can be identified by a deduplication system for a block size recognized by the deduplication system). Given a technique to generate a data stream that is not deduplicatable, other techniques can then be used to generate a data stream with a specific level of deduplication. In certain testing scenarios the specification of a specific deduplication level is very desirable.

In some embodiments, a data stream is generated at a high speed of approximately 2.3+GB per second on 64-bit machines with a single CPU. In some embodiments, a data stream is 100% reproducible on any computer. In some embodiments, a data stream does not repeat any block for 500 TB or more. In some embodiments, four billion or more unique data streams can be generated. In some embodiments, a data stream is unique from any other data stream generated from a different initialization parameter and/or a different pair of prime numbers. Furthermore, any block of a data stream is different from any block of any other data stream generated from a different initialization parameter and/or a different pair of prime numbers.

Embodiments of verifying a data stream without a master copy of the data stream or the parameters used to generate the data stream are described herein. In various embodiments, the parameters used to generate the data stream include at least the initialization parameter, the first prime number, and the second prime number. Whereas conventionally, a master copy (e.g., an original copy of the data stream that is used as a reference copy) of a data stream is required to perform verification of the data stream (e.g., as restored from a storage), as will be described in further detail below, a data stream as generated by embodiments described herein can be verified without a master copy of the data stream and/or even the parameters used to generate the data stream.

Embodiments of generating data that is not compressible are described herein. In various embodiments, “non-compressible” data refers to data that cannot be compressed (e.g., using common compression techniques). Data that cannot be compressed refers to data whose size remains unchanged or even increases (e.g., by an amount of overhead data generated by a compression technique) after the data is subjected to a compression technique. As will be described in further detail below, data can be non-compressible due to certain characteristics of the values in the data. For example, because compression techniques exploit redundancy in data, non-compressible data is generated in a manner that minimizes the redundancy among its values. In various embodiments, a “non-compressible sequence” refers to a sequence of values generated using an initialization parameter and a constrained prime number. In various embodiments, a “non-compressible data stream” refers to a sequence of values generated by merging two non-compressible sequences, each of which is generated using the same initialization parameter and a respective constrained prime number. As will be described in further detail below, a “constrained” prime number refers to a prime number that meets a predetermined set of criteria and therefore can be used to generate non-compressible data. In various embodiments, an “unconstrained” prime number refers to a prime number that does not meet a predetermined set of criteria and therefore may not be used to generate non-compressible data. In some embodiments, a set of constrained prime numbers is first identified. In some embodiments, a non-compressible first sequence associated with a first constrained prime number and the initialization parameter is obtained. In some embodiments, a non-compressible second sequence associated with a second constrained prime number and the initialization parameter is obtained. In some embodiments, the first constrained prime number and the second constrained prime number are selected based on a revision parameter that is received. For example, a revision parameter can map to two constrained prime numbers (or in some cases, two unconstrained prime numbers or a constrained prime number and an unconstrained prime number). A non-compressible data stream is generated by merging (e.g., interleaving) the first non-compressible sequence and the second non-compressible sequence associated with the respective first and second constrained prime numbers. In some embodiments, a non-compressible data stream is not deduplicatable.

FIG. 1 is a diagram showing an embodiment of a system for performing quality assurance on a storage duplication server. In the example, system 100 includes storage deduplication server 102, storage device 108, network 104, and quality assurance server 106. Network 104 includes high-speed data networks and/or telecommunication networks. Storage device 108 comprises a single storage device such as a hard disk, a tape drive, a semiconductor memory, a plurality of storage devices such as a redundant array system (e.g., a redundant array of independent disks (RAID)), a system for storage such as a library system or network attached storage system, or any other appropriate storage device or system.

System 100 includes a quality assurance environment in which quality assurance server 106 generates data streams that are sent over network 104 to storage deduplication server 102. Storage deduplication server 102 is configured to deduplicate data that it receives from quality assurance server 106 with respect to data that is already stored at storage device 108. Storage device 108 is attached to or otherwise accessible by storage deduplication server 102. For example, storage deduplication server 102 comprises a backup server that is configured to store at storage device 108 backup data received from a source location (e.g., quality assurance server 106). In some embodiments, storage deduplication server 102 is configured to segment each received data stream into data blocks (e.g., of a fixed size or of variable sizes) and perform deduplication with respect to each data block. For example, a data stream comprises a sequence of values and each data block comprises one or more values. In various embodiments, “deduplicating a data block” refers to determining whether the data block to be stored has already been stored at a target location (e.g., storage deduplication server 102 and/or storage device 108). In the event that the data block has not already been stored at the target location, the data block is stored at the target location (e.g., by storage deduplication server 102). Otherwise, in the event that the data block has already been stored at the target location, a reference, pointer, link, and/or other associating data to the previously stored data block is stored at the target location (e.g., by storage deduplication server 102) instead of another instance of the data block. In various embodiments, a reference, pointer, link, and/or other associating data to the stored data block comprises a relatively smaller amount of data relative to the amount of data associated with the data block. When a data stream stored at storage device 108 is to be restored (e.g., at the source location), the stored data blocks and/or references to stored data blocks associated with the data stream can be used to reconstruct the data stream. Deduplication can be used to reduce the amount of data that is stored at the target location by only storing new (non-duplicate) data that has not yet been stored at the target location and by storing references to data that has been previously stored at the target location.

For example, the parameters used in the generation of data streams and/or other attributes associated with the data streams are at least known to if not also controlled by quality assurance server 106. How storage deduplication server 102 performs deduplication with respect to storing at least two data streams that it receives from quality assurance server 106 given known data such as, for example, the percentage of difference in data between the two data streams, may indicate a deduplication result (e.g., a quality and/or effectiveness) of the storage deduplication techniques used by storage deduplication server 102. The deduplication result can be used to determine whether the deduplication techniques used by storage deduplication server 102 should be reconfigured, for example.

In some embodiments, quality assurance server 106 is configured to generate a data stream based on parameters such as an initialization parameter (sometimes referred to as a “seed value”) and two prime numbers selected from a constrained modified set of prime numbers. In some embodiments, each pair of two prime numbers to be used with the initialization parameter (seed value) is selected based on a received revision parameter (sometimes referred to as a “revision value” and where two different revision values with respect to the same initialization parameter each maps to a different pair of prime numbers). A sequence is determined for the initialization parameter and each of the two prime numbers. In various embodiments, a data stream is determined by merging (e.g., interleaving) the two sequences. For example, merging the two sequences comprises creating a new (merged) sequence that includes each value from the first sequence followed by a corresponding value from the second sequence. For example, a first value from the first sequence that corresponds to a second value from the second sequence is associated with the same position within the first sequence as the second value is within the second sequence (e.g., a first value in position 1 of the first sequence corresponds to a second value in position 1 of the second sequence).

In some embodiments, if the two selected prime numbers each meets a predetermined set of criteria (as will be described in further detail below), then the two prime numbers comprise constrained prime numbers. A non-compressible sequence is generated with the same initialization parameter and each of the two constrained prime numbers and the two non-compressible sequences can be merged (e.g., interleaved) to generate a non-compressible data stream. Otherwise, if the two selected prime numbers each do not meet a predetermined set of criteria, then the two prime numbers comprise unconstrained prime numbers. A compressible sequence is generated with the same initialization parameter and each of the two unconstrained prime numbers and the two compressible sequences can be merged (e.g., interleaved) to generate a compressible data stream.

In some embodiments, this generated data stream is sent by quality assurance server 106 over network 104 to storage deduplication server 102 (e.g., as part of a test backup operation) for storage. Storage deduplication server 102 is configured to segment the data stream into data blocks (e.g., of fixed or variable sizes) and store only the new data blocks (e.g., data blocks that have not already been stored at storage device 108). If, for example, in a test backup operation, none of the data blocks of the data stream have already been stored at storage device 108, storage deduplication server 102 will store all the data blocks of the data stream. If, for example, in a test backup operation, some of the data blocks of the data stream have already been stored at storage device 108, storage deduplication server 102 will store references in place of the data blocks that have already been stored and store all the remaining data blocks of the data stream. Because the data stream is not deduplicatable, storage deduplication server 102 will not identify any duplicate data blocks within the data stream.

After the data stream is stored by storage deduplication server 102, the stored data stream may be restored. For example, restoring a stored data stream includes reconstructing the data stream using the stored data blocks and/or references to stored data blocks associated with the data stream. To test the accuracy or reliability of the storage deduplication techniques and/or the restoration techniques used by storage deduplication server 102, in some embodiments, quality assurance server 106 is configured to verify the data stream that was stored by and thereafter restored by the storage deduplication server 102. In some embodiments, quality assurance server 106 is configured to verify the correctness of the restored data stream by comparing the restored data stream to the original data stream that quality assurance server 106 had generated and then sent to storage deduplication server 102. As will be described in further detail below, in various embodiments, the restored data stream itself can be used to verify its correctness without requiring a master copy of the original data stream and/or the parameters used to generate the original data stream, thereby eliminating the need to maintain a master copy of the data stream for verification purposes. In various embodiments, a data stream can be verified in the same manner regardless if the data stream is compressible or non-compressible.

System 100 shows one example in which embodiments of data stream generation as described herein can be applied. Data stream generation may be applied in various other applications, as appropriate.

FIG. 2 is a diagram showing an embodiment of a quality assurance server. In some embodiments, quality assurance server 106 of system 100 of FIG. 1 can be implemented using the example of FIG. 2. The quality assurance server of FIG. 2 includes parameter engine 202, sequence generator engine 204, data stream generator engine 206, constrained prime number identification engine 212, verification engine 207, and local storage 210. Each of parameter engine 202, sequence generator engine 204, data stream generator engine 206, constrained prime number identification engine 212, and verification engine 207 can be implemented using one or both of software and hardware. Local storage 210 comprises a local storage or a networked file system storage.

Parameter engine 202 is configured to provide parameters to use to generate a data stream. In various embodiments, parameters to use to generate a data stream include at least an initialization parameter and a revision parameter. In some embodiments, an initialization parameter is a seed value. In some embodiments, the seed value is any value selected from an address space that is represented by N bits (e.g., the address space comprising (0, . . . , 2^(N)−1)). N can be selected to be any positive integer. For example, if N=5, then 2^(N-5)=32 so the address space is (0, . . . , 31) and the initialization parameter can be selected to be any value from (0, . . . , 31). In some embodiments, a revision parameter is a revision value associated with a given “seed value” that uniquely maps to at least two prime numbers. In some embodiments, each of the at least two prime numbers is selected using the revision parameter from a set of prime numbers that is modified to exclude “2” and include “1” and is also constrained/bounded by 2^(N)−1. In some embodiments, each of the at least two prime numbers is selected using the revision parameter from a set of constrained prime numbers (e.g., that is identified by constrained prime number identification engine 212, as described below).

In some embodiments, one or more of the initialization parameter (the seed value) and the revision parameter (the revision value), which maps to two or more prime numbers, are input by a user (e.g., associated with performing quality assurance). In some embodiments, one or more of the initialization parameter and the revision parameter, which maps to two or more prime numbers, are generated by a computer program.

In various embodiments, parameter engine 202 is configured to provide the initialization parameter and the revision parameter to sequence generator engine 204.

Constrained prime number identification engine 212 is configured to identify constrained prime numbers that are to be used to generate non-compressible sequences and data streams. In various embodiments, a “constrained” prime number refers to a prime number that meets a predetermined set of criteria and is therefore usable with the initialization parameter to generate a non-compressible sequence (e.g., by sequence generator engine 204). Two or more non-compressible sequences, each generated using the same initialization parameter and a corresponding constrained prime number, can be merged (e.g., interleaved) to generate a non-compressible data stream. Whereas prime numbers that do not meet the predetermined set of criteria (“unconstrained” prime numbers) can be used to generate sequences and such sequences can be merged together to form data streams, neither such sequences nor such data streams are associated with the property of being non-compressible. In other words, sequences and/or data streams generated with prime numbers that do not meet the predetermined set of criteria (“unconstrained” prime numbers) may be compressible.

In some embodiments, constrained prime number identification engine 212 is configured to iterate through a set of numbers and determine whether each number meets the predetermined set of criteria. Those numbers of the set that meet the predetermined set of criteria are included in an identified set of constrained prime numbers. In some embodiments, the set of numbers through which constrained prime number identification engine 212 iterates comprises an address space represented by N bits (e.g., the address space comprising (0, . . . , 2^(N)−1)). N can be selected to be any positive integer.

As will be described in further detail below, the predetermined set of criteria that is used to identify constrained prime numbers requires that a constrained prime number 1) comprises a prime number from the set of prime numbers that is modified to exclude “2” and include “1,” 2) includes component values that are each individually prime numbers from the set of prime numbers that is modified to exclude “2” and include “1,” and 3) includes no duplicate component values. As such, the set of constrained prime numbers comprises a subset of all the prime numbers in the set of prime numbers that is modified to exclude “2” and include “1.” In various embodiments, a “component value” of a number represents a subset of bits included in the number. For example, where a number is represented by N=32 bits, the 32-bit number p can be represented by a sequence of four bytes (each byte includes 8 bits), p3p2p1p0. In the example of a 32-bit number p that includes four bytes, each byte is referred to as a “component value.”

Sequence generator engine 204 is configured to receive the initialization parameter and the revision parameter from parameter engine 202 to use to generate at least two sequences. In some embodiments, sequence generator engine 204 is configured to generate a sequence using each pair of the initialization parameter and a prime number selected using the revision parameter received from parameter engine 202. An example technique by which to generate each such sequence is described in more detail below. For example, if the revision parameter that was received from parameter engine 202 maps to two prime numbers, then sequence generator engine 204 will generate two corresponding sequences. Similarly, if the revision parameter that was received from parameter engine 202 maps to three prime numbers, then sequence generator engine 204 will generate three corresponding sequences. In some embodiments, if sequence generator engine 204 uses a constrained prime number to generate a sequence, then the sequence will be non-compressible. In some embodiments, if sequence generator engine 204 uses an unconstrained prime number (a prime number that is not from the set of constrained prime numbers) to generate a sequence, then the sequence may be compressible. In some embodiments, sequence generator engine 204 is configured to send the generated sequences and/or the corresponding given set of the initialization parameter and the revision parameter received from parameter engine 202 to be stored at local storage 210. In some embodiments, sequence generator engine 204 is configured to send the generated sequences and/or the corresponding given set of the initialization parameter and the revision parameter to data stream generator engine 206 for data stream generator engine 206 to use to generate a data stream.

Data stream generator engine 206 is configured to receive the at least two sequences and/or the corresponding given set of the initialization parameter and the revision parameter from sequence generator engine 204. In some embodiments, data stream generator engine 206 is configured to merge the at least two sequences into one new (merged) sequence that serves as the generated data stream. In some embodiments, if data stream generator engine 206 merges two sequences that were each generated using a constrained prime number, then the generated data stream will be non-compressible. In some embodiments if data stream generator engine 206 merges two sequences that were each generated using an unconstrained prime number (a prime number that is not from the identified set of constrained prime numbers), then the generated data stream may be compressible. In some embodiments, the at least two sequences are merged into one sequence by creating a new merged sequence in which each value from the first sequence is followed by a corresponding value from each other sequence (i.e., the two sequences are interleaved). For example, of the two sequences that are to be merged to become the data stream, the first sequence comprises {S11, S12, S13, . . . } and the second sequence comprises {S21, S22, S23, . . . }. In this example, merging the first and second sequences will yield the following data stream {S11, S21, S12, S22, S13, S23, . . . }. In some embodiments, data stream generator engine 206 is configured to send the generated data stream and/or the corresponding given set of the initialization parameter and the revision parameter to store at local storage 210. In some embodiments, data stream generator engine 206 is configured to send the generated data stream to an external destination (e.g., storage deduplication server 102 of system 100 of FIG. 1).

Verification engine 207 is configured to receive a data stream and verify the data stream without another (e.g., a master) copy of the data stream or the parameters (e.g., the initialization parameter, the first prime number, and the second prime number) that were used to generate the data stream. For example, the data stream to be verified is data restored from a storage device (e.g., by storage deduplication server 102 of system 100 of FIG. 1). It may be desirable to verify the received data stream to determine that the values of the data stream correctly match the pattern of values associated with a merging (e.g., interleaving) of two (or more) sequences generated by an initialization parameter and (at least) two prime numbers even if none of the initialization parameter and two prime numbers are known/retrieved prior to the start of the verification process. In some embodiments, verification engine 207 is configured to use a portion of the data stream to deduce the values of the first prime number and the second prime number and then use the first prime number and the second prime number to verify at least a portion of the data stream. In various embodiments, a data stream can be verified in the same manner regardless if the data stream is compressible or non-compressible. For example, verifying the data stream includes determining whether the difference between every other value of the data stream alternately equals the first prime number and the second prime number. For example, if the data stream can be successfully verified, then the techniques used to restore the data stream from the storage device can be determined to be effective. Otherwise, if the data stream cannot be successfully verified, then the techniques used to restore the data stream from the storage device can be determined to be ineffective and reconfiguration is needed.

FIG. 3 is a flow diagram showing an embodiment of a process of generating a sequence using an initialization parameter and a prime number. In some embodiments, process 300 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 300 is implemented at quality assurance server 106 of system 100 of FIG. 1.

Parameters may be provided to use to generate a data stream. Such parameters include an initialization parameter (e.g., a starting value or seed value) and a revision parameter (e.g., a revision value). The revision value maps to or is used to select at least two prime numbers from a constrained modified set of prime numbers. Process 300 can be performed to generate a sequence for each pair of the initialization parameter and a prime number (selected using the revision parameter). For example, if an initialization parameter (seed) and a revision parameter that maps to two prime numbers (prime1 and prime2) were received, then process 300 can be performed twice: once to generate a first sequence using the seed and prime1 and a second time to generate a second sequence using the seed and prime2. For example, the first and second sequences can be used to generate a data stream using another process (e.g., process 400 of FIG. 4, below).

At 302, an initialization parameter and a prime number are received, wherein the prime number is selected from a constrained modified set of prime numbers. For example, the initialization parameter can be received from a user input or from a computer program. In some embodiments, the initialization parameter comprises a seed value that is selected from an address space (0, . . . , 2^(N)−1), where N is selected to be any positive integer (e.g., 32). In some embodiments, the prime number is selected by/mapped to by a received revision parameter (e.g., a revision value associated with the seed value). The prime number is selected from a modified set of prime numbers that excludes “2” but includes “1” and that is bounded/constrained by 2^(N)−1.

At 304, a sequence is generated based at least in part on the initialization parameter and the prime number. In some embodiments, the sequence is of length 2^(N). In various embodiments, the first value of the sequence is the initialization parameter (starting value or seed value). Each subsequent value of the sequence is determined as a function of the prior value in the sequence, the prime number, and 2^(N). For example, each subsequent value of the sequence is determined as the sum of the prior value in the sequence and the prime number and then the sum modulo 2^(N). In some embodiments, sequences generated using the same initialization parameter but different prime numbers will not have any blocks of values (e.g., 8 KiB in size) in common with each other.

An example of generating a sequence using an initialization parameter (seed value) and a prime number is described below:

Below are some definitions that will be used by the following examples:

prime: Any natural prime number

Prime: Set of natural prime numbers (2, 3, 5, 7, . . . )

PrimeM: A set of prime numbers that excludes 2 (even though “2” is considered a prime number) from the set Prime and includes 1 (even though “1” is not considered a prime number)

primeM: A member of the set PrimeM

PrimeN: Set of prime numbers that are less than 2^(N)−1

PrimeMN: Set of primeM numbers that are less than 2^(N)−1

Example sets of prime numbers:

-   -   Prime5=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]     -   PrimeM5=[1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]     -   Prime6=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,         53, 59, 61]     -   PrimeM6=[1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,         53, 59, 61]

Regarding the Set PrimeMN

For a given address space (0, . . . , 2^(N)−1) and a given a number s (seed value) in the chosen address space, if any specific number from the PrimeMN set is added to seed value s (with modulo-2^(N)), then the resulting sequence will repeat only after all numbers in the address space have been visited. This is not true for the set PrimeN as this property will not hold for the prime number 2. However, this property also holds for the number 1. That is the reason for excluding 2 and including 1 to the set PrimeMN. In some embodiments, set PrimeMN is sometimes referred to as a “constrained modified set of prime numbers.”

Below is an example of generating a sequence:

Address space is N=5 bits (so the address space includes (0, . . . , 31)),

PrimeM5 is set [1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31],

The selected seed value (e.g., the initialization parameter) (which is any number between 0 . . . 31 inclusive) is 14,

The selected primeM5 is 3.

The resulting sequence S(N, seed, prime) will be as follows:

S(5, 14, 3)=

14, 17, 20, 23, 26, 29, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 2, 5, 8, 11

The following table, Table 1, illustrates that the values of sequence S(5, 14, 3) are obtained by using the seed value (14) as the first value of the sequence and obtaining each subsequent value in the sequence by incrementing the previous value in the sequence by the prime number (3) with modulo 2^(N-5)=32:

TABLE 1 Start +3 +3 +3 +3 +3 +3 +3 +3 14 17 20 23 26 29 0 3 Next +3 6 9 12 15 18 21 24 27 Next +3 30 1 4 7 10 13 16 19 Next +3 22 25 28 31 2 5 8 11 End

This holds true for any selected member from the set PrimeM5.

In the above example, if the seed value was changed to 10 then the resulting sequence will be as follows:

S(5, 10, 3)=

10, 13, 16, 19, 22, 25, 28, 31, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 1, 4, 7

The following table, Table 2, illustrates that the values of sequence S(5, 10, 3) are obtained by using the seed value (10) as the first value of the sequence and obtaining each subsequent value in the sequence by incrementing the previous value in the sequence by the prime number (3) with modulo 2^(N-5)=32:

TABLE 2 Start +3 +3 +3 +3 +3 +3 +3 +3 10 13 16 19 22 25 28 31 Next +3 2 5 8 11 14 17 20 23 Next +3 26 29 0 3 6 9 12 /5 Next +3 18 21 24 27 30 1 4 7 End

The sequences S(5, 14, 3) and S(5, 10, 3) are basically the same except for the rotation in the positions of their respective values. The values of sequence S(5, 14, 3) can be found starting from value 14 of sequence S(5, 10, 3), which is underlined in Table 2, above.

However, for example, if each value were represented by 32 bits, then if a different prime number is selected, then the resulting sequence will not have any blocks (e.g., blocks that are 8 KiB in size) of values in common with any other sequence.

In another example where the seed value is 14 and the selected primeM5 is 5:

S(5, 14, 5)=

14, 19, 24, 29, 2, 7, 12, 17, 22, 27, 0, 5, 10, 15, 20, 25, 30, 3, 8, 13, 18, 23, 28, 1, 6,11,16,21,26,31,4,9

The following table, Table 3, illustrates that the values of sequence S(5, 14, 5) are obtained by setting the first value to the seed value (14) and obtaining each subsequent value in the sequence by incrementing the previous value in the sequence by the prime number (5) with modulo 2^(N=5)=32:

TABLE 3 Start +5 +5 +5 +5 +5 +5 +5 +5 14 19 24 29 2 7 12 17 Next +5 22 27 0 5 10 15 20 25 Next +5 30 3 8 13 18 23 28 1 Next +5 6 11 16 21 26 31 4 9 End

In the above example, if the seed value was changed to 10 then the resulting sequence will be as follows:

S(5, 10, 5)=

10, 15, 20, 25, 30, 3, 8, 13, 18, 23, 28, 1, 6, 11, 16, 21, 26, 31, 4, 9, 14, 19, 24, 29, 2, 7, 12, 17, 22, 27, 0, 5

The following table, Table 4, illustrates that the values of sequence S(5, 10, 5) are obtained by setting the first value to the seed value (10) and obtaining each subsequent value in the sequence by incrementing the previous value in the sequence by the prime number (5) with modulo 2^(N=5)=32:

TABLE 4 Start +5 +5 +5 +5 +5 +5 +5 +5 10 15 20 25 30 3 8 13 Next +5 18 23 28 1 6 11 16 21 Next +5 26 31 4 9 14 19 24 29 Next +5 2 7 12 17 22 27 0 5 End

The sequences S(5, 14, 5) and S(5, 10, 5) are basically the same except for the rotation in the positions of their respective values. The values of sequence S(5, 14, 5) can be found starting from value 14 of sequence S(5, 10, 5), which is underlined in Table 4.

However, sequence S(5, 14, 5) does not have any (e.g., 8 KiB) blocks of (e.g., 32-bit) values in common with the sequence S(5, 14, 3) or any other sequence S(5, seed, prime) when the prime is any number other than 5.

The technique described above to generate a sequence is an example and other techniques may be used to generate a sequence using a seed value and a prime number from a constrained modified set of prime numbers.

FIG. 4 is a flow diagram showing an embodiment of a process of generating a data stream using prime numbers. In some embodiments, process 400 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 400 is implemented at quality assurance server 106 of system 100 of FIG. 1.

At 402, an initialization parameter is received. For example, the initialization parameter can be received from a user input or from a computer program. In some embodiments, the initialization parameter comprises a seed value that is selected from an address space (0, . . . , 2^(N)−1), where N is selected to be any positive integer (e.g., 32).

At 404, a first sequence associated with a first prime number and the initialization parameter is obtained. In some embodiments, each of two prime numbers is selected from a modified set of prime numbers that excludes “2” but includes “1” and that is bounded/constrained by 2^(N)−1. The two prime numbers may comprise the same prime number or different prime numbers. In some embodiments, the two prime numbers are selected based on a received revision parameter comprising a revision value. In some embodiments, a first sequence is generated using the initialization parameter and one of the two prime numbers using a process such as process 300 of FIG. 3. In some embodiments, the first sequence is received from another entity.

At 406, a second sequence associated with a second prime number and the initialization parameter is obtained. In some embodiments, a second sequence is generated using the initialization parameter and the prime number of the two prime numbers that was not used to generate the first sequence of step 404. In some embodiments, the second sequence is generated using the initialization parameter and the prime number of the two prime numbers that was not used to generate the first sequence using a process such as process 300 of FIG. 3. In some embodiments, the second sequence is received from another entity.

At 408, a data stream is generated including by merging the first sequence and the second sequence. In some embodiments, merging the first sequence and the second sequence includes interleaving the first and second sequences into a new sequence that is referred to as the data stream. In some embodiments, the data stream includes a sequence of alternating values from the first and second sequences. For example, if two sequences are to be merged to become the data stream, and the first sequence comprises {S11, S12, S13, . . . } and the second sequence comprises {S21, S22, S23, . . . }, then merging the first and second sequences will yield the following data stream {S11, S21, S12, S22, S13, S23, . . . }. In some embodiments, the data stream is not deduplicatable.

In some embodiments, more than two prime numbers from the constrained modified set of prime numbers can be selected based on the revision value and the data stream can be generated by merging more than two sequences, each of which is generated using the initialization parameter and a respective prime number.

An example of generating a data stream using an initialization parameter (seed value) and at least two prime numbers is described below:

Two or more S sequences, as described above, can be used to build a new merged sequence (data stream).

For example:

MS2 is a merged sequence (data stream) of two S sequences, and has the following four parameters:

N Address space (0, . . . , 2^(N)−1),

seed The seed value (e.g., the initialization parameter) (the first value) of each of sequence #1 and sequence #2,

prime1 A first prime number selected from set PrimeMN for sequence S(N, seed, prime1),

prime2 A second prime number selected from set PrimeMN for sequence S(N, seed, prime2).

MS2(N, seed, prime1, prime2)=S(N, seed, prime1)+S(N, seed, prime2)

Similarly, MS3 is a merged sequence (data stream) of three S sequences, sequence #1, sequence #2, and sequence #3, that has five parameters:

N Address space (0, . . . , 2^(N)−1),

seed The seed value (e.g., the initialization parameter) (the first value) of each of sequence #1 and sequence #2,

prime1 A first prime number selected from set PrimeMN for sequence S(N, seed, prime1),

prime2 A second prime number selected from set PrimeMN for sequence S(N, seed, prime2),

prime3 A third prime number selected from set PrimeMN for sequence S(N, seed, prime3).

MS3 (N, seed, prime1, prime2, prime3)=

S(N, seed, prime1)+S(N, seed, prime2)+S(N, seed, prime3)

For example, given N=5, seed=14, prime1=3 and prime2=5, the sequence MS2 (5, 14, 3, 5) (data stream) is produced by alternately inserting one value from each individual sequence S(5, 14, 3) and S(5, 14, 5) into a merged sequence.

MS2(5, 14, 3, 5)=

14, 14, 17, 19, 20, 24, 23, 29, 26, 2, 29, 7, 0, 12, 3, 17, 6, 22, 9, 27, 12, 0, 15, 5, 18, 10, 21, 15, 24, 20, 27, 25, 30, 30, 1, 3, 4, 8, 7, 13, 10, 18, 13, 23, 16, 28, 19, 1, 22, 6, 25, 11, 28, 16, 31, 21, 2, 26, 5, 31, 8, 4, 11, 9

The following table, Table 5, illustrates that the values of data stream MS2(5, 14, 3, 5) are obtained by alternately inserting a value from sequence S(5, 14, 3) and a value from sequence S(5, 14, 5) (the values of each sequence are individually displayed with Table 1 and Table 3, above) into a merged sequence:

TABLE 5 Start Start +3 +5 +3 +5 +3 +5 14 14 17 19 20 24 23 29 26 2 29 7 0 12 3 17 6 22 9 27 12 0 15 5 18 10 21 15 24 20 27 25 30 30 1 3 4 8 7 13 10 18 13 23 16 28 19 1 22 6 25 11 28 16 31 21 2 26 5 31 8 4 11 9

Note in Table 5 above, the values from sequence S(5, 14, 5) are shown in italics while the values from sequence S(5, 14, 3) are not shown in italics.

In another example, given N=5, seed=10, prime1=3 and prime2=5, the sequence MS2(5, 10, 3, 5) is produced by alternately inserting one value from each of individual sequences S(5, 10, 3) and S(5, 10, 5) into a merged sequence.

MS2(5, 10, 3, 5)=

10, 10, 13, 15, 16, 20, 19, 25, 22, 30, 25, 3, 28, 8, 31, 13, 2, 18, 5, 23, 8, 28, 11, 1, 14, 6, 17, 11, 20, 16, 23, 21, 26, 26, 29, 31, 0, 4, 3, 9, 6, 14, 9, 19, 12, 24, 15, 29, 18, 2, 21, 7, 24, 12, 27, 17, 30, 22, 1, 27, 4, 0, 7, 5

The following table, Table 6, illustrates that the values of data stream MS2(5, 10, 3, 5) are obtained by alternately inserting a value from sequence S(5, 10, 3) and a value from sequence S(5, 10, 5) (the values of each sequence are individually displayed with Table 2 and Table 4, above) into a merged sequence:

TABLE 6 Start Start +3 +5 +3 +5 +3 +5 10 10 13 15 16 20 19 25 22 30 25 3 28 8 31 13 2 18 5 23 8 28 11 1 14 6 17 11 20 16 23 21 26 26 29 31 0 4 3 9 6 14 9 19 12 24 15 29 18 2 21 7 24 12 27 17 30 22 1 27 4 0 7 5

Note in Table 6 above, the values from sequence S(5, 10, 5) are shown in italics while the values from sequence S(5, 10, 3) are not shown in italics.

It was shown above that sequences S(N, seed1, prime) and S(N, seed2, prime) included the same values except for the rotation in the positions of their respective values. This does not hold true for the data stream, MS2. As shown with the two examples of data stream MS2, each seed value produces an entirely new data stream. Note that the pattern/consecutive values of 14 and 14 appear only in data stream MS2(5, 14, 3, 5) and not in merged sequence data stream MS2(5, 10, 3, 5).

Using the merging of two sequences as described above, 2^(N) data streams (one for each seed value in the address space) can be generated for any given pair of prime numbers (prime1, prime2). Each data stream determined from merging two sequences has 2*2^(N) values.

Assume that N=32 (the most often used size in bits of an unsigned integer) then the length of the data stream MS2(32, seed, prime1, prime2) will be as follows:

2*(2³²) unsigned integers of 32 bit size or 4*2*(2³²) bytes≈32 gigabytes (GiB).

As such, a data stream MS2(32, seed, prime1, prime2) will repeat after exactly 32 GiB. Put another way, each seed value will provide a new data stream and if N=32, then 2³² seed values are available to use to build approximately 4 billion data streams where each data stream will be exactly 32 GiB long.

In some embodiments, each revision value (e.g., the revision parameter) with respect to a given seed value uniquely maps to a first prime number of a fixed value and a second prime number that is associated with a position within the constrained modified set of prime numbers that matches the revision value. For example, each pair of prime numbers can be represented by (prime1, prime2). For example, given seed=10; revision 0 can map to the pair of prime1=3 and prime2=1, revision 1 can map to the pair of prime1=3 and prime2=3, revision 2 can map to the pair of prime1=3 and prime2=5, revision 4 can map to the pair of prime1=3 and prime2=7, and so forth.

In some embodiments, each revision value (which maps to a different pair of prime numbers (prime1, prime2)) for a given seed value can be used to generate a data stream that is distinct from any data stream that is generated from the same seed value and any other revision value. In some embodiments, each revision value (which maps to a different pair of prime numbers (prime1, prime2)) for a given seed value can be used to generate a data stream that is distinct from any data stream that is generated from any other seed value and revision value.

Because for a given seed value, merged sequence data stream MS2(32, seed, prime1, prime2) will repeat after approximately 32 GiB, if the desired application of the data stream requires a data stream to be longer than 32 GiB, then one or more enhancements can be performed to increase the length of the data stream.

Below are some example enhancements that can be performed to increase the length of a data stream:

Enhancement #1:

MS2 comprises two simple sequences S(N, seed, prime1) and S(N, seed, prime2).

This enhancement automatically alters the prime used for the first sequence (prime1) to a new value when the repetition is about to occur.

This enhancement allows for a very large sequence to be built as long as we have a prime number available.

Enhancement #2:

Let each value of a sequence be represented by 64 bits. Therefore, let N=64 (instead of N=32). Where N=64, a generated data stream will not repeat for approximately 2*(2^(N-64)) unsigned integers of 64 bit size or 8*2*(2⁶⁴) bytes≈256 exabytes.

Enhancement #3:

More than two sequences are merged together to generate a data stream.

For example:

A merged sequence data stream that includes three sequences can be represented as MS3(N, seed, prime1, prime2, prime3).

Depending on the merging mode utilized, this can produce extremely long sequences.

The technique described above to generate a data stream is an example and other techniques may be used to generate a data stream using a seed value and at least two prime numbers selected from a constrained modified set of prime numbers.

In some embodiments, as described above, each seed value can be used to generate data streams that are distinct from data streams generated with any other seed values and a given seed value with a revision value can be used to generate a data stream that is distinct from a data stream generated with the given seed value and any other revision value. Therefore, in certain applications, different seed values and/or revision values can be assigned to different users involved in performing quality assurance such that each group of users and/each user in a group can perform testing on their respective data stream (generated with a given seed value and revision value) without impacting the testing performed by any other user. For example, each group of users (e.g., in a quality assurance team) can be assigned a seed value and each user within a group can be assigned a revision value with respect to that group's assigned seed value so each user within the group can use their respectively assigned seed value and revision value to generate a data stream distinct from each other's.

In some embodiments, a data stream comprising the merging (e.g., interleaving) of two (or more) sequences can be generated in memory using a small memory footprint. Below is pseudocode that describes one example implementation of a data stream generation engine as described in some embodiments:

The following four variables can be created and stored in memory:

prime1

prime2

accumulator)

accumulator1

The inputs to the data stream generation engine are seed (e.g., the initialization parameter), prime1 (e.g., a first prime number), and prime2 (e.g., a second prime number). The variables are initialized using the input parameters:

Set accumulator1=seed

Set accumulator2=seed

Set accumulator1=accumulator1+prime1

Set accumulator2=accumulator2+prime2

As will be shown in further detail below, accumulator1 represents the values from a first sequence generated using seed and prime1 and accumulator2 represents the values from a second sequence generated using seed and prime2.

The values of the data stream are generated by alternately outputting a value from each of the two sequences. As such, in the pseudocode below, the data stream is generated by alternately outputting a value from each of accumulator1 and accumulator2 and modifying both accumulator1 and accumulator2 after outputting from accumulator1 and accumulator2. The below pseudocode for outputting the values of the data stream can be repeated until a stop condition is met (e.g., the stop condition can be that either of the output of accumulator1 or accumulator2 is the same as a value previously output by accumulator1 or accumulator2, respectively, which indicates that the values of the data stream are starting to repeat).

Output accumulator1

Output accumulator2

Set accumulator1=accumulator1+prime1

Set accumulator2=accumulator2+prime2

The data stream generation as described in the above example implementation is extremely fast because only two ADD operations are used.

Embodiments of verifying a data stream are described herein. In various embodiments, a data stream generated in accordance with the embodiments described above can be verified without another copy (e.g., a master copy) of the data stream and without the parameters that were used to generate the data stream (e.g., the initialization parameter, a first prime number, a second prime number). A verification capability that does not require another copy (e.g., a master copy) of the data stream for comparison purposes is invaluable for proving the correctness of a storage system. For example, a verification capability that does not require another copy of the data stream for comparison purposes (or even the parameters that were used to generate the data stream) can free up storage space that would have otherwise been used to store the other copy of the data stream and/or the parameters. In various embodiments, verification can be performed on any part of a data stream. A self-verification capability is valuable for proving the store and restore capabilities of a storage system.

A data stream may be verified for correctness in various different applications. For example, to verify a data stream for correctness is to confirm that the data stream is actually the merging (e.g., interleaving) of two sequences (e.g., each of which is generated using a process such as process 300 of FIG. 3, above). In a first example application, data stored at a storage device associated with a data stream is restored and the restored version of the data stream can be verified to test the effectiveness of storing and/or restoring techniques. In a second example application, a newly generated data stream can be verified to confirm that the data stream had been correctly generated.

FIG. 5 is a flow diagram showing an embodiment of a process of verifying a data stream. In some embodiments, process 500 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 500 is implemented at quality assurance server 106 of system 100 of FIG. 1.

In some embodiments, process 500 describes an example of performing verification on a data stream that was generated by merging (e.g., interleaving) two sequences (e.g., using a process such as process 400 of FIG. 4). In some embodiments, process 500 describes an example of performing verification on a non-compressible data stream that was generated by merging (e.g., interleaving) two non-compressible sequences (e.g., using a process such as process 1200 of FIG. 12, below). As will be described in further detail below, process 500 deduces the two (e.g., unconstrained or constrained) prime numbers associated with the respective two sequences and uses these two prime numbers to verify the (e.g., compressible or non-compressible) data stream.

At 502, a data stream is received. For example, the data stream is restored from data stored at a storage device (e.g., by a quality assurance server such as quality assurance server 106 of system 100 of FIG. 1). In another example, the data stream is recently generated (e.g., by a quality assurance server such as quality assurance server 106 of system 100 of FIG. 1).

At 504, a first prime number is determined based at least in part on a difference between a first pair of non-consecutive values from the data stream. Each of the two sequences that were interleaved to generate the data stream is based on a prime number and the initialization parameter (e.g., seed value). Each of the two sequences initially starts with the seed value plus the prime number associated with that sequence and each subsequent value is generated by a prior value plus the prime number associated with that sequence. As such, the difference between every other value of the data stream should equal one of the two prime numbers associated with the sequences that were interleaved to form the data stream. For example, a first prime number can be deduced as the difference between a pair of values of the data stream that are separated by a value (e.g., the Xth and (X+2)th values of the data stream).

At 506, a second prime number is determined based at least in part on a difference between a second pair of non-consecutive values from the data stream. Similarly, a second prime number can be deduced as the difference between another pair of values of the data streams that are separated by a value (e.g., the (X+1)th and (X+3)th values of the data stream).

At 508, the first prime number and the second prime number are used to verify the data stream. Once the first and second prime numbers have been deduced, the data stream (or any portion thereof) can be verified based on determining whether the difference between pairs of values separated by a value of the data stream matches one of the first and second prime numbers. In some embodiments, while the initialization parameter comprising a seed value was used to generate the data stream, the seed value is not used in verifying the data stream and therefore does not need to be determined.

In some embodiments, a data stream consists of two interleaved sequences each based upon their own prime number resulting in a data stream that is not deduplicatable. For example, each sequence starts with the seed value (seed) plus their individual prime (prime1 or prime2) and subsequent values are generated by the prior value plus their individual prime. Therefore, the values in a data stream in some embodiments are (where value[X] represents the value in position X in the data stream):

seed (in hexadecimal)=0E00000E, prime1 (in hexadecimal)=0103050D, prime2 (in hexadecimal)=0305070B

value1=seed+prime1 0F03051B=0E00000E+0103050D

value2=seed+prime2 11050719=0E00000E+0305070B

value3=value1+prime1 10060A28=0F03051B+0103050D

value4=value2+prime2 140A0E24=11050719+0305070B

value5=value3+prime1 11090F35=10060A28+0103050D

value6=value4+prime2 170F152F=140A0E24+0305070B

value7=value5+prime1 120C1442=11090F35+0103050D

value8=value6+prime2 1A141C3A=170F152F+0305070B

In this data stream, the first value and every other value are from the sequence generated with prime1 (the values above written in bold) and the second value and every other value are from the sequence with prime2 (the values written not in bold).

The difference of the Xth and (X+2)th values is either prime1 if the Xth value was from the first sequence or prime2 if the Xth value was from the second sequence.

seed (in hexadecimal)=0E00000E, prime1 (in hexadecimal)=0103050D, prime2 (in hexadecimal)=0305070B

value3−value1=prime1 10060A28−0F03051B=0103050D

value4−value2=prime2 140A0E24−11050719=0305070B

value5−value3=prime1 11090F35−10060A28=0103050D

value6−value4=prime2 170F152F−140A0E24=0305070B

value7−value5=prime1 120C1442−11090F35=0103050D

value8−value6=prime2 1A141C3A−170F152F=0305070B

As shown above, four consecutive values of the data stream are enough to determine the two prime numbers (prime1 and prime2). Once the values of prime1 and prime2 are deduced, the correctness of the entire data stream can be established as all subsequent values must be equal to the prior value plus an alternating prime1 and prime2 value.

For verification, the input can be at least a portion from the start or middle of the data stream. In various embodiments, values for prime1 and prime2 can be deduced and the entire data stream verified as long as a minimum of four values of the data stream are made available.

The following is pseudocode that shows the example steps that will cause the portion of the data stream comprising value3, value4, value5, value6, value7 and value8 to be verified. Note: The data stream started with value1 but the data stream verification is being attempted from value3.

Input: value3

save value3 in accumulator1

Input: value4

save value4 in accumulator2

Input: value5

save difference of value5 and accumulator1 in prime1

save value5 in accumulator1

Input: value6

save difference of value6 and accumulator2 in prime2

save value6 in accumulator2

At this point, the prime1 and prime2 values are deduced.

Input: value7

ensure that the difference of value7 and accumulator 1 equals prime1

save value7 in accumulator 1

Input: value8

ensure that the difference of value8 and accumulator2 equals prime2

save value8 in accumulator2

As shown above, in some embodiments, prime1 and prime2 associated with a data stream to be verified can be derived just from four (e.g., initial) values of the data stream. In some embodiments, the seed value can also be derived using the (e.g., initial) four values and the values for the prime1 and prime2. In various embodiments, the seed value is not required for verification but can be deduced and reported (e.g., to enable the regeneration of the exact same data stream if desired).

If prime1 is repeatedly added to the first value of the data stream and prime2 is repeatedly added to the second value of the data stream and when both the accumulators are equal, the seed is found.

Since the initial value of each interleaved sequence is value=seed+prime and all succeeding values are value=value+prime, given enough additions of prime to value, value will at some point be equal to seed due to modulo arithmetic wrap around.

Therefore, the seed can be deduced by repeatedly performing the following additions, described in pseudocode, as required:

value_from_prime1_sequence=value_from_prime1_sequence+prime1

value_from_prime2 sequence=value_from_prime2_sequence+prime2

Until value_from_prime1 sequence=value_from_prime2 sequence. The two values will be equal only when they are both equal to the seed value, seed.

In some embodiments, a data stream comprising the merging (e.g., interleaving) of two (or more) sequences can be verified in memory using a small memory footprint. Below is pseudocode that describes one example implementation of a data stream verification engine as described in some embodiments:

The following six variables can be created and stored in memory:

prime1

prime2

accumulator)

accumulator2

next_step=Initialization#1

result=true

In step Initialization#1, accumulator 1 is initialized by inputting a first value of the data stream to be used in the verification process:

Set accumulator1=value

Set next_step=Initialization#2

In step Initialization#2, accumulator2 is initialized by inputting a next value of the data stream:

Set accumulator2=value

Set next_step=Initialization#3

In step Initialization#3, the difference between a next value of the data stream and accumulator 1 is set as prime1:

Set prime1=value−accumulator 1

Set accumulator1=value

Set next_step=Initialization#4

In step Initialization#4, the difference between a next value of the data stream and accumulator2 is set as prime2:

Set prime2=value−accumulator2

Set accumulator2=value

Set next step=Verify#1

In step Verify#1, it is checked whether the difference between the next value of the data stream and accumulator1 equals prime1:

if (value−accumulator1) does not equal prime1 then set result to false

Set accumulator1=value

Set next step as Verify#2

In step Verify#2, it is checked whether the difference between the next value of the data stream and accumulator2 equals prime2:

if (value−accumulator2) does not equal prime2 then set result to false

Set accumulator2=value

Set next step as Verify#1

Verify#1 and Verify#2 are alternately performed until a stop condition is met (e.g., the end of the data stream has been reached). If result is ever set to false, then the data stream cannot be verified to be correct. However, if result remains set to true after the stop condition is met, then the data stream is verified to be correct.

The data stream verification as described in the above example implementation is extremely fast because only a few subtraction operations are used.

Embodiments of generating data that is not compressible are described herein. A data stream used for testing may be first compressed. If the data stream is compressible, then the size of the compressed data set (plus the overhead data associated with compression) will most likely be smaller than the size of the original data stream. However, in certain testing scenarios, it may be desirable to preserve the (approximate) size of the data stream even if the data stream undergoes a compression process so as to better observe the effectiveness of a separate process that is applied to the data stream. Given a method to generate a data stream that is not compressible, other methods can then be used to generate a data stream with a specific level of compression. In certain testing scenarios the specification of a specific compression level is very desirable.

For example, a 2 GiB data stream may be compressed into a 1.5 GiB data stream prior to a test deduplication process. In the test deduplication process, the test data stream is to be compared against previously stored data of which 0.5 GiB is known to be common to the uncompressed 2 GiB data stream. However, if the compressed 1.5 GiB data stream is compared to the previously stored data in the deduplication process and 0.4 GiB of the compressed 1.5 GiB data stream is determined by the deduplication process to be common to the previously stored data, then because of the effect of compression on the tested data stream, it is unclear whether the 0.1 GiB discrepancy in the deduplication is a result of a fault in the deduplication process or the compression technique. As such, in some embodiments, a data stream is generated to be non-compressible so that a particular testing technique (e.g., of deduplication) with the data stream can be isolated from any compression techniques that may be applied to the data stream.

In some embodiments, deduplication systems deal with data in “blocks.” A deduplication system can use either fixed or variable sized blocks. An example of variable block sizes is block sizes ranging from 4 KiB to 12 KiB with an average size of 8 KiB. The data streams can be segmented at certain natural boundaries and variable sized blocks are created. The blocks that are duplicates (of previously stored data) are detected and only the unique blocks are stored in the deduplication storage. Instead of storing a duplicate block multiple times, a reference to the previously stored block is stored. The reference requires significantly less storage space than the duplicate data blocks would have required. In the event that the deduplication process occurs at a client that is remote from the server associated with the deduplication storage, detection of duplicate blocks results in significantly less network bandwidth than sending the actual block data from the client to the server. Requiring less storage space and less network bandwidth is traded for requiring more CPU cycles for duplicate block detection, duplicate block reference storage, and lookup and duplicate block retrieval.

Many compression techniques deal with data at the bit and byte level. Compression techniques typically replace occurrences of often repeated series of bytes in the data with a reference or code value that is smaller than the often repeated sequence of bits or bytes. For example, replacing frequent multiple occurrences of a repeated series of three byte values with a one byte code throughout a data set reduces the size of a data set, which then requires less storage to store the data set. A compressor builds a dictionary of smaller sequences on-the-fly that are used to replace larger sequences. A compressed data set also requires less network bandwidth to transfer. Requiring less storage space to store a data set and less network bandwidth to transfer a data set is traded for requiring CPU cycles for compression and decompression.

In some embodiments, deduplication systems perform both duplicate data set block detection and data set compression. Data blocks can be compressed before or after duplicate block detection.

In some embodiments, each value in a sequence or in a data stream, which comprises merging two or more sequences, can be represented in hexadecimal. Examples of using hexadecimal to represent values are described below:

Note that all commas and underscores used below are merely for easier reading of the digits of a number.

Computers store all values in binary (base 2). Each additional bit represents another power factor of 2. One bit (2¹=2 values) can store the values 0, 1 in decimal and 0, 1 in binary. Two bits (2²=4 values) can store the values 0, 1, 2, 3 in decimal or 00, 01, 10, 11 in binary. Four bits (2⁴=16 values) can store the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 in decimal or 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 in binary. We now have four bits described. Four bits is called a “nibble.” Using the same math, eight bits (2⁸=256 values) can store the values 0 to 255 in decimal or 0000_0000 to 1111_1111 in binary. Eight bits is called a “byte” and consists of two nibbles. If a number were represented by N bits, the highest value storable is 2^(N)−1 bits.

Referring back to the four bit (2⁴=16 values) example, the four bits can be perceived as a single digit or a two digit decimal number from 0 to 15, or a single four digit binary number from 0000 to 1111. For convenience, base 16 is also used to represent the sixteen values of a four bit nibble using the single hexadecimal digits 0 to 9 and A to F. The A to F hexadecimal digits can also be in lower case. The sixteen four bit values in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. In order to distinguish a hexadecimal number from a decimal number, it is generally written with a prefix of “0x” or a suffix of “hexadecimal” or “hex.”

Similarly, for convenience, an eight bit number can also be perceived as two four bit nibbles. Since a single hexadecimal digit represents each four bit nibble value, two hexadecimal digits represent two four bit nibbles. Together, two four bit nibbles represent a full eight bit byte value. In other words, an eight bit byte can be represented by two hexadecimal digits, where each digit can be one of 0 to 9 or A to F.

Example eight bit (byte) values in decimal, binary, and hexadecimal representations:

0 decimal, 0000_0000 binary, 00 hex (or 0x00)

10 decimal, 0000_1010 binary, 0A hex (or 0x0A)

50 decimal, 0011_0010 binary, 32 hex (or 0x32)

203 decimal, 1100_1011 binary, CB hex (or 0xCB)

255 decimal, 1111_1111 binary, FF hex (or 0xFF)

If 32 bits were used to represent a value, the largest decimal value that can be represented is 2³²−1=4,294,967,295. As such, 32 bits can be used to represent just over 4,000,000,000 decimal or 4 billion values, 2³²=4,294,967,296, ranging from 0 to 4,294,967,295. For convenience, a 32-bit number is frequently perceived as four sets of eight bits or four bytes. Each byte's value is written using two base 16 digits. There is one hexadecimal digit for each nibble in a byte. This is more convenient because with practice, one can “see” the individual bits by looking at the four bit base 16 nibbles and their digits. It is also easier to remember and write A028FC1D instead of 2687040541.

An example 32-bit value in binary, decimal and hexadecimal representations:

1010_0000_0010_1000_1111_1100_0001_1101 binary (very hard to remember, all bits visible)

A0_28_FC_1D hex (easier to remember and the underlying four bit sets can be seen)

2,687,040,541 decimal (around 2.687 billion) (harder to remember, bit sets obscured)

In some embodiments, a data stream is generated as described in some embodiments with 32-bit values, at least some of which may include component eight bit or byte values that can repeat and therefore be replaced by a shorter sequence of bits or bytes. As such, such data streams are compressible. A compressor can locate and replace larger sequences of byte values that repeat within the data stream with a smaller sequence of bits or bytes, thereby reducing the storage required to store the data stream.

Below are some examples of 32-bit values in decimal representation and their byte values in hexadecimal:

17 decimal—00_00_00_11 hex

3,439,329,280 decimal—CD_00_00_00 hex

2,818,572,318 decimal—A8_00_00_1E hex

31,247 decimal—00_00_7A_0F hex

287,456,068 decimal—11_22_3B_44 hex

4,179,770,180 decimal—F9_22_3B_44 hex

A compressor may locate the repeated multiple byte sequences 00_00 or 00_00_00 or 22_3B_44 across the different 32-bit values in the data stream.

In various embodiments, a prime number, primeM (a member of set of prime numbers PrimeM that includes “1” and excludes “2” and from an address space defined by (0, . . . , 2^(N)−1)) that meets a predetermined set of criteria is specially identified as a “constrained” prime number, which can be used to generate non-compressible sequences and non-compressible data streams, which comprise a merging of multiple such sequences. In some embodiments, the predetermined set of criteria for a “constrained” N-bit prime number comprises 1) that a constrained prime number comprises a prime number from a set of prime numbers that is modified to exclude “2” and include “1” and is also constrained/bounded by 2^(N)−1 2) that each component value of the constrained prime number comprises a prime number from the set of prime numbers that is modified to exclude “2” and include “1” and 3) that none of the component values of the constrained prime number are duplicates. In some embodiments, a constrained prime number is represented by N bits and each component value of the constrained prime number comprises a subset of the values (e.g., a byte) of the constrained prime number.

In actual implementation, the number of bits to use to represent a value, N, can be selected to be of various values. For purposes of illustration, in various examples below, N is selected to be 32 bits. A constrained 32-bit prime number comprises four component eight bit byte values that are each a prime number in the range 0 to 2⁸−1 (=255) (including “1” and excluding “2”) and that no byte prime number is a duplicate within the same 32-bit prime number. Put another way, the component byte values of a prime number p of N=32 bits can be represented as p3p2p1p0 and p would be identified as a constrained prime number if each of p0, p1, p2, and p3 is a prime number in the range 0 to 255 (including “1” and excluding “2”) and none of p0, p1, p2, and p3 were duplicates. This avoids the case where bytes in a data stream were not uniformly distributed, which most of the time is for lower values of primes. Therefore, a compressor can no longer locate any repeated multiple byte sequences in the sequence or data stream.

The following table shows all primes that are members of PrimeM8 (PrimeM8 is a modified set of prime numbers that excludes “2” but includes “1” and that is bounded/constrained by 2⁸−1=255) set:

PrimeM8=[1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,

67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,

157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251]

In hexadecimal representation, PrimeM8=[01, 03, 05, 07, 0B, 0D, 11, 13, 17,

1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB]

Where N=32 bits, any member of the set PrimeM32 (PrimeM32 is a modified set of prime numbers that excludes “2” but includes “1” and that is bounded/constrained by 2³²−1=4,294,967,295) that has individual component byte values outside of the PrimeM8 members are not constrained prime numbers and therefore not considered for non-compressible data generation.

Any member of the set PrimeM32 that has duplicate individual component byte values (PrimeM8 members) are also not constrained prime numbers and therefore not considered for non-compressible data generation.

The component byte values in an unconstrained primeM number can be any value from 0 to 255 dec or 00 to FF hex. One or more of the four component byte values (PrimeM8 members) within an unconstrained 32-bit value can be the same (duplicate). This results in generated data that is compressible. Below are three example 32-bit values (with four component byte values) represented as decimal and hexadecimal that do not meet the three predetermined criteria for constrained prime numbers and are therefore unconstrained prime numbers:

A) 11 decimal, 00_00_00_0B hex

B) 16,777,729 decimal, 01_00_02_01 hex

C) 16,777,751 decimal, 01_00_02_17 hex

In the example above, the hexadecimal representation of value A includes duplicate “00” byte values and also non-prime value “00,” the hexadecimal representation of value B includes duplicate “01” byte values and also non-prime value “00,” and the hexadecimal representation of value C includes non-prime value “00.” Therefore, each of values A, B, and C are compressible.

Below are two example 32-bit values (with four component byte values) represented as decimal and hexadecimal that do meet the three predetermined criteria for constrained prime numbers:

D) 16,975,117 decimal, 01_03_05_0D hex

E) 50,661,131 decimal, 03_05_07_0B hex

In the example above, the hexadecimal representations of values D and E are each a prime number, the four component byte values of each are prime numbers, and none of the four component byte values of each of values D and E are duplicates. Therefore, each of values D and E are not compressible.

Generally, the component byte values in a constrained primeM number must each be a primeM number in the range 0 to 255 decimal or 00 to FF hex, and all four component byte values must have a different value.

FIG. 6 is a diagram showing a table that includes sample constrained and unconstrained 32-bit primeM values to help explain how a primeM value is determined to be constrained or unconstrained. In FIG. 6, a primeM value that is “good” is a constrained prime number that will result in generated data that cannot be compressed and a primeM value that is “bad” is an unconstrained prime number that may result in generated data that may be compressed.

FIG. 7 is a flow diagram showing an embodiment of a process for identifying a set of constrained prime numbers. In some embodiments, process 700 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 700 is implemented at quality assurance server 106 of system 100 of FIG. 1.

Process 700 is an example process of identifying a set of constrained prime numbers from a set of numbers constrained by N bits by iterating through each number of the set and determining whether the number meets a predetermined set of criteria associated with steps 704, 706, and 708 required for a constrained prime number. As described herein, a constrained prime number can be used with an initialization parameter to generate a non-compressible sequence.

At 702, a (next) number from a constrained set of numbers is obtained. For example, if N=32 bits, then the constrained set of numbers includes (0, . . . , 2³²−1).

At 704, it is determined whether the number is a prime number. In various embodiments, the prime number is from a set of prime numbers that is modified to exclude “2” and include “1” and is also constrained/bounded by 2^(N)−1. In the event that the number is a prime number, control is transferred to 706. Otherwise, in the event that the number is not a prime number, control is transferred to 712.

At 706, it is determined whether each component value of the number is itself a prime number. In some embodiments, the size of each component value of the number is a byte (eight bits). For example, if N=32 bits and the size of each component value is a byte, then each number p of the constrained set would have four component byte values p3p2p1p0. For example, each component byte value of p3p2p1p0 should be a prime number from a set of prime numbers that is modified to exclude “2” and include “1” and within range 0 to 255 (set Prime8). In some embodiments, the size of each component value of the number is dependent on the size with which a particular compression technique determines compression. In the event that the each component value of the number is a prime number, control is transferred to 708. Otherwise, in the event that not every component value of the number is a prime number, control is transferred to 712.

At 708, it is determined whether there is a duplicate component value in the number. There should not be any duplicates among the component values of a constrained prime number. Put another way, each component value of a constrained prime number must be different from each other. Returning to the former example, if N=32 bits and the size of each component value is a byte, then there should be no duplicate component values among the four component byte values p3p2p1p0 of number p. In the event that there are no duplicate component values in the number, control is transferred to 710. Otherwise, in the event that there are duplicate component values in the number, control is transferred to 712.

At 710, the number is included in a set of constrained prime numbers. If the number meets the three criteria of steps 704, 706, and 708, then the number is determined to be a constrained prime number and included in a set of constrained prime numbers.

At 712, it is determined whether there is at least one more number in the constrained set of numbers. In the event that there is at least one more number in the constrained set of numbers, control is returned 702. Otherwise, in the event that there are no more numbers in the constrained set of numbers, process 700 ends.

FIG. 8 is a flow diagram showing an embodiment of a process for generating a non-compressible sequence using an initialized parameter and a constrained prime number. In some embodiments, process 800 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 800 is implemented at quality assurance server 106 of system 100 of FIG. 1.

At 802, an initialization parameter is received. For example, the initialization parameter can be received from a user input or from a computer program. In some embodiments, the initialization parameter comprises a seed value that is selected from an address space (0, . . . , 2^(N)−1), where N is selected to be any positive integer (e.g., 32).

At 804, a constrained prime number is determined, wherein the constrained prime number comprises a plurality of component values, wherein each of the plurality of component values comprises a prime number, wherein each of the plurality of component values is different. In some embodiments, a constrained prime number is selected from the identified set of constrained prime numbers based on a received revision parameter (e.g., associated with the initialization parameter). In some embodiments, a constrained prime number is selected from the identified set of constrained prime numbers based on any appropriate technique. In some embodiments, a set of constrained prime numbers can be identified using a process such as process 700 of FIG. 7.

At 806, a non-compressible sequence is generated based at least in part on the initialization parameter and the constrained prime number. In some embodiments, the initial value of the non-compressible sequence comprises the sum of the seed value and the selected constrained prime number and each subsequent value comprises the sum of the prior value and the constrained prime number.

For example, the sequence generator engine generates the initial 32-bit value of the sequence by computing:

Accumulator=seed+primeM

where seed represents the initialization parameter and primeM represents the selected constrained prime number. The sequence generator engine always generates the next 32-bit value of the sequence by computing:

Accumulator=Accumulator+primeM

Please note that the numbers in hexadecimal are written with underscores merely to help visualize the value of the four individual bytes (eight bits each) that are each a component value of a 32-bit (four byte) primeM number.

FIG. 9A is a diagram showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number. In FIG. 9A, the initialization parameter (“seed”) and the unconstrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “01000201,” respectively. The values of the sequence of FIG. 9A are ordered from left-to-right in each row from the top to the bottom. In FIG. 9A, the initial value of the sequence (“0F00020F”) is a sum of the seed and the unconstrained prime number and each subsequent value is the sum of the prior value and the unconstrained prime number. Prime number “01000201” is not a constrained prime number because it includes duplicate component byte values of “00” and also a non-prime component byte value of “00.” Because an unconstrained prime number was used to generate the sequence, the sequence includes values that can be compressed. Therefore, the sequence generated with an unconstrained prime number is compressible.

As shown in the table of FIG. 9A, within each 32-bit value of the sequence, there can be component byte values that are the same. Some byte values occur much more frequently than other byte values.

FIG. 9B is a diagram showing a frequency analysis table for all component byte values of a sampling of 1,032 bytes of the sequence of FIG. 9A. The frequency values of FIG. 9B are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The least significant hexadecimal nibble values are located across the top of the table. The generated data sequence that was shown in FIG. 9A is compressible since the frequency distribution of all 256 possible byte values is not uniformly distributed.

FIG. 10A is a diagram showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number. In FIG. 10A, the initialization parameter (“seed”) and the constrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “0103050D,” respectively. The values of the sequence of FIG. 10A are ordered from left-to-right in each row from the top to the bottom. In FIG. 10A, the initial value of the sequence (“0F03051B”) is a sum of the seed and the constrained prime number and each subsequent value is the sum of the prior value and the constrained prime number. Prime number “0103050D” is a constrained prime number because each component byte value is a prime number and there are no duplicate component byte values. Because a constrained prime number was used to generate the sequence, the sequence includes values that cannot be compressed. Therefore, the sequence generated with a constrained prime number is non-compressible.

As shown in the table of FIG. 10A, within each 32-bit value of the sequence, there are rarely component values that are the same. There are no repeated 32-bit values. All the component byte values of the non-compressible sequence of FIG. 10A occur with basically the same frequency. The thirty-six data values in lines #11 through #16 marked by vertical bars are extracted and used in the table of FIG. 11, below.

FIG. 10B is a diagram showing a frequency analysis table for all component byte values of a sampling of 1,032 bytes of the sequence of FIG. 10A. The frequency values of FIG. 10B are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The least significant hexadecimal nibble values are located across the top of the table. The generated data is not compressible since the frequency distribution of all 256 possible byte values in this sampling of 1032 bytes is near uniformly distributed. Even when the sample size is decreased to 516 bytes or increased to 2064 bytes or higher, the distribution stays near uniform.

FIG. 10C is a diagram showing a frequency analysis table for all component byte values of a sampling of 12,288 bytes of the sequence of FIG. 10A. The frequency values of FIG. 10C are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The least significant hexadecimal nibble values are located across the top of the table. Even though FIG. 10C shows a frequency table with a greater sampling size of the sequence of FIG. 10A than that used in the frequency table of FIG. 10B, the table of FIG. 10C shows that the generated data is not compressible since the frequency distribution of all 256 possible byte values in this sampling of 12,288 bytes is still near uniformly distributed.

FIG. 11 is a diagram showing a table of accumulator (generated data) internal byte value rotations. The thirty-six 32-bit data values used for the table of FIG. 11 are taken from the six marked data lines #11 through #16 in the table of FIG. 10A. The initialization parameter (“seed”) used was “0E00000E.” The constrained prime number (“primeM”) used was “0103050D.” Its four component byte values are p3 p2 p1 p0 in bytes b3, b2, b1, b0. All initial component byte values are themselves a prime number (in decimal): 1, 3, 5 and 13. Each time the “accumulator=accumulator+primeM” step is performed, the accumulator bytes increment and rotate plus any applicable carries on each addition. The component byte values in each column visit the 256 values in the eight bit address space as the accumulator value visits the four billion values in the 32-bit address space.

FIG. 12 is a flow diagram showing an embodiment of a process for generating a non-compressible data stream using two constrained prime numbers. In some embodiments, process 1200 is implemented at system 100 of FIG. 1. Specifically, in some embodiments, process 1200 is implemented at quality assurance server 106 of system 100 of FIG. 1.

Process 1200 shows an example process of generating a non-compressible data stream by merging (e.g., interleaving) two sequences, each generated using the same initialization parameter (e.g., a seed value) and a respective unconstrained prime number.

At 1202, an initialization parameter is received. For example, the initialization parameter can be received from a user input or from a computer program. In some embodiments, the initialization parameter comprises a seed value that is selected from an address space (0, . . . , 2^(N)−1), where N is selected to be any positive integer (e.g., 32).

At 1204, a first non-compressible sequence associated with a first constrained prime number and the initialization parameter is obtained. In some embodiments, each of two constrained prime numbers is selected from an identified set of constrained prime numbers. In some embodiments, the set of constrained prime numbers is identified using a process such as process 700 of FIG. 7. The two constrained prime numbers may comprise the same constrained prime number or different constrained prime numbers. In some embodiments, the two constrained prime numbers are selected based on a received revision parameter comprising a revision value. In some embodiments, a first non-compressible sequence is generated using the initialization parameter and one of the two constrained prime numbers using a process such as process 800 of FIG. 8. In some embodiments, the first non-compressible sequence is received from another entity.

At 1206, a second non-compressible sequence associated with a second constrained prime number and the initialization parameter is obtained. In some embodiments, a second non-compressible sequence is generated using the initialization parameter and the constrained prime number of the two constrained prime numbers that was not used to generate the first non-compressible sequence of step 1204. In some embodiments, the second non-compressible sequence is generated using the initialization parameter and the constrained prime number of the two prime numbers that was not used to generate the first non-compressible sequence using a process such as process 800 of FIG. 8. In some embodiments, the second non-compressible sequence is received from another entity.

At 1208, a non-compressible data stream is generated including by merging the first non-compressible sequence and the second non-compressible sequence. In some embodiments, a data stream having the property of being non-compressible is generated including by merging the first non-compressible sequence and the second non-compressible sequence. In some embodiments, merging the first non-compressible sequence and the second non-compressible sequence includes interleaving the first and second non-compressible sequences into a new sequence that is referred to as the non-compressible data stream. In some embodiments, the data stream includes a sequence of alternating values from the first and second non-compressible sequences. For example, of two sequences that are to be merged to become the data stream, the first sequence comprises {S11, S12, S13, . . . } and the second sequence comprises {S21, S22, S23, . . . }, then merging the first and second sequences will yield the following data stream {S11, S21, S12, S22, S13 S23, . . . }. In some embodiments, the non-compressible data stream is also not deduplicatable.

In some embodiments, more than two constrained prime numbers from the constrained modified set of prime numbers can be selected based on the revision value and the non-compressible data stream can be generated by merging more than two non-compressible sequences, each of which is generated using the initialization parameter and a respective constrained prime number.

In some embodiments, a non-compressible data stream generated using a process such as process 1200 can be verified using a process such as process 500 of FIG. 5.

FIGS. 13A to 15H illustrate examples of two non-compressible sequences that can be merged to generate a non-compressible data stream. FIGS. 13A to 13E are diagrams that illustrate the values and properties of a first non-compressible sequence that is generated with an initialization parameter (e.g., a seed value) and a first constrained prime number. FIGS. 14A to 14E are diagrams that illustrate the values and properties of a second non-compressible sequence that is generated with the initialization parameter (e.g., a seed value) and a second constrained prime number. FIGS. 15A to 15H are diagrams that illustrate the values and properties of a non-compressible data stream that is generated by merging the two non-compressible sequences respectively associated with FIGS. 13A to 13E and FIGS. 14A to 14E.

FIGS. 13A to 13D are diagrams showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number. In FIGS. 13A to 13D, the initialization parameter (“seed”) and the constrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “0103050D,” respectively. The values of the sequence span FIGS. 13A to 13D and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 13E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 13A to 13D. The frequency values of FIG. 13E are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 19. The lowest frequency count is 13. The delta is 6 between the highest and lowest frequency counts. The generated data is not compressible since the frequency distribution of all 256 possible byte values is roughly uniformly distributed.

The uncompressed sequence of FIGS. 13A to 13D has a file size of 4,080 bytes. After compressing the sequence of FIGS. 13A to 13D (e.g., using a known ZIP compression technique), the compressed file size is 4,133 bytes. The data generated using constrained primes is not compressible so therefore the compressed file size is larger than the uncompressed file size due to the compression metadata overhead bytes.

FIGS. 14A to 14D are diagrams showing a table that includes 32-bit values of a non-compressible sequence generated using an initialization parameter and a constrained prime number. In FIGS. 14A to 14D, the initialization parameter (“seed”) and the constrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “0305070B,” respectively. The values of the sequence span FIGS. 14A to 14D and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 14E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 14A to 14D. The frequency values of FIG. 14E are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 20. The lowest frequency count is 12. The delta is 8 between the highest and lowest frequency counts. The generated data is not compressible since the frequency distribution of all 256 possible byte values is roughly uniformly distributed.

The uncompressed sequence of FIGS. 14A to 14D has a file size of 4,080 bytes. After compressing the sequence of FIGS. 14A to 14D (e.g., using a known ZIP compression technique), the compressed file size is 4,133 bytes. The data generated using constrained primes is not compressible so therefore the compressed file size is larger than the uncompressed file size due to the compression metadata overhead bytes.

FIGS. 15A to 15G are diagrams showing a table that includes 32-bit values of a non-compressible data stream generated from merging two non-compressible sequences. In FIGS. 15A to 15G, the initialization parameter (“seed”), the first constrained prime number (“primeM1”) used to generate the first sequence (described with FIGS. 13A to 13D, above), and the second constrained prime number (“primeM2”) used to generate the second sequence (described with FIGS. 14A to 14D, above) are represented in hexadecimal as “0E00000E,” “0103050D,” and “0305070B,” respectively. The values of the data stream span FIGS. 15A to 15G and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 15H is a diagram showing a frequency analysis table for all component byte values of 8,124 bytes of the non-compressible data stream of FIGS. 15A to 15G. The frequency values of FIG. 15H are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 37. The lowest frequency count is 28. The delta is 9 between the highest and lowest frequency counts. The generated data is not compressible since the frequency distribution of all 256 possible byte values is roughly uniformly distributed.

The uncompressed data stream of FIGS. 15A to 15G has a file size of 8,184 bytes. After compressing the sequence of FIGS. 15A to 15G (e.g., using a known ZIP compression technique), the compressed file size is 8,241 bytes. The data generated using constrained primes is not compressible so therefore the compressed file size is larger than the uncompressed file size due to the compression metadata overhead bytes.

In contrast to the examples of FIGS. 13A to 15H, FIGS. 16A to 18H illustrate examples of two compressible sequences that can be merged to generate a compressible data stream. FIGS. 16A to 16E are diagrams that illustrate the values and properties of a first compressible sequence that is generated with an initialization parameter (e.g., a seed value) and a first unconstrained prime number. FIGS. 17A to 17E are diagrams that illustrate the values and properties of a second compressible sequence that is generated with the initialization parameter (e.g., a seed value) and a second unconstrained prime number. FIGS. 18A to 18H are diagrams that illustrate the values and properties of a compressible data stream that is generated by merging the two compressible sequences respectively associated with FIGS. 16A to 16E and FIGS. 17A to 17E, respectively.

FIGS. 16A to 16D are diagrams showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number. In FIGS. 16A to 16D, the initialization parameter (“seed”) and the unconstrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “01000201,” respectively. The values of the sequence span FIGS. 16A to 16D and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 16E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 16A to 16D. The frequency values of FIG. 16E are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 140. The lowest frequency count is 10. The delta is 130 between the highest and lowest frequency counts. The generated data is compressible since the frequency distribution of all 256 possible byte values is not uniformly distributed.

The uncompressed sequence of FIGS. 16A to 16D has a file size of 4,080 bytes. After compressing the sequence of FIGS. 16A to 16D (e.g., using a known ZIP compression technique), the compressed file size is 3,934 bytes. The data generated using unconstrained primes is compressible so therefore the uncompressed file size is larger than the compressed file size and the number of bytes removed due to compression exceeds the compression metadata overhead bytes.

FIGS. 17A to 17D are diagrams showing a table that includes 32-bit values of a compressible sequence generated using an initialization parameter and an unconstrained prime number. In FIGS. 17A to 17D, the initialization parameter (“seed”) and the unconstrained prime number (“primeM”) used to generate the sequence are represented in hexadecimal as “0E00000E” and “00000017,” respectively. The values of the sequence span FIGS. 17A to 17D and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 17E is a diagram showing a frequency analysis table for all component byte values of 4,080 bytes of the sequence of FIGS. 17A to 17D. The frequency values of FIG. 17E are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 1,034. The lowest frequency count is 3. The delta is 1,031 between the highest and lowest frequency counts. The generated data is compressible since the frequency distribution of all 256 possible byte values is not uniformly distributed.

The uncompressed sequence of FIGS. 17A to 17D has a file size of 4,080 bytes. After compressing the sequence of FIGS. 17A to 17D (e.g., using a known ZIP compression technique), the compressed file size is 1,621 bytes. The data generated using unconstrained primes is compressible so therefore the uncompressed file size is larger than the compressed file size and the number of bytes removed due to compression exceeds the compression metadata overhead bytes.

FIGS. 18A to 18G are diagrams showing a table that includes 32-bit values of a compressible data stream generated from merging two compressible sequences. In FIGS. 18A to 18G, the initialization parameter (“seed”), the first unconstrained prime number (“primeM1”) used to generate the first sequence (described with FIGS. 16A to 16D, above), and the second unconstrained prime number (“primeM2”) used to generate the second sequence (described with FIGS. 17A to 17D, above) are represented in hexadecimal as “0E00000E,” “01000201,” and “00000017,” respectively. The values of the data stream span FIGS. 18A to 18G and are ordered from left-to-right in each row from the top to the bottom of each figure.

FIG. 18H is a diagram showing a frequency analysis table for all component byte values of 8,184 bytes of the compressible data stream of FIGS. 18A to 18G. The frequency values of FIG. 18H are in decimal representation. The most significant hexadecimal nibble values are located down the left of the table. The highest frequency count is 1,176. The lowest frequency count is 15. The delta is 1,161 between the highest and lowest frequency counts. The generated data is compressible since the frequency distribution of all 256 possible byte values is not uniformly distributed.

The uncompressed data stream of FIGS. 18A to 18G has a file size of 8,184 bytes. After compressing the sequence of FIGS. 18A to 18G (e.g., using a known ZIP compression technique), the compressed file size is 6,852 bytes. The data generated using unconstrained primes is compressible so therefore the uncompressed file size is larger than the compressed file size and the number of bytes removed due to compression exceeds the compression metadata overhead bytes.

Although the foregoing embodiments have been described in some detail for purposes of clarity of understanding, the invention is not limited to the details provided. There are many alternative ways of implementing the invention. The disclosed embodiments are illustrative and not restrictive. 

What is claimed is:
 1. A system, comprising: a processor configured to: receive an initialization parameter; determine a first constrained prime number and a second constrained prime number, wherein a constrained prime number comprises a plurality of component values, wherein each of the plurality of component values comprises a prime number, wherein each of the plurality of component values is different; and generate a non-compressible data stream comprising a plurality of data blocks at least in part by: generate a first non-compressible sequence based at least in part on the initialization parameter and the first constrained prime number; generate a second non-compressible sequence based at least in part on the initialization parameter and the second constrained prime number, wherein the first and second non-compressible sequences are each comprised of a non-repeating sequence of numbers; and merge the first non-compressible sequence and the second non-compressible sequence to generate the non-compressible data stream comprising the plurality of data blocks, wherein values of the non-compressible data stream alternate between values of the first non-compressible sequence and values of the second non-compressible sequence; and send the non-compressible data stream comprising the plurality of data blocks to be stored at a storage device, wherein the storage device is configured to store all of the plurality of data blocks in response to determining that none of the data blocks have been already stored at the storage device; receive restored data associated with the non-compressible data stream from the storage device, wherein the restored data associated with the non-compressible data stream comprises the plurality of data blocks; determine the first constrained prime number based at least in part on a difference between a first pair of non-consecutive values from the restored data associated with the non-compressible data stream; determine the second constrained prime number based at least in part on a difference between a second pair of non-consecutive values from the restored data associated with the non-compressible data stream; and verify an accuracy and/or reliability of the storage device, wherein to verify the accuracy and/or reliability of the storage device, the restored data associated with the non-compressible data stream is verified by using the determined first constrained prime number and the determined second constrained prime number; and a memory coupled to the processor and configured to provide the processor with instructions.
 2. The system of claim 1, wherein each of the plurality of component values comprises a byte value.
 3. The system of claim 1, wherein each of the plurality of component values comprises a prime number from a set of prime numbers including “1” and excluding “2” in a range of 0 to
 255. 4. The system of claim 1, wherein the constrained prime number comprises a prime number from a set of prime numbers including “1” and excluding “2” in a range of 0 to 2^(N)−1, where N is a positive integer.
 5. The system of claim 1, wherein the constrained prime number is determined from an identified set of constrained prime numbers based at least in part on a revision parameter.
 6. The system of claim 1, wherein the restored data is verified without a master copy of the non-compressible data stream.
 7. The system of claim 1, wherein to verify the restored data associated with the non-compressible data stream by using the first constrained prime number and the second constrained prime number comprises: determine of a first offset between the first pair of non-consecutive numbers from the restored data; determine of a second offset between the second pair of non-consecutive numbers from the restored data; and verify the data stream in response to determining the first offset is equal to the first constrained prime number and the second offset is equal to the second constrained prime number.
 8. A method, comprising: receiving an initialization parameter; determining a first constrained prime number and a second constrained prime number, wherein a constrained prime number comprises a plurality of component values, wherein each of the plurality of component values comprises a prime number, wherein each of the plurality of component values is different; generating a non-compressible data stream comprising a plurality of data blocks at least in part by: generating, by a processor, first a non-compressible sequence based at least in part on the initialization parameter and the first constrained prime number; generating, by the processor, a second non-compressible sequence based at least in part on the initialization parameter and the second constrained prime number, wherein the first and second non-compressible sequences are each comprised of a non-repeating sequence of numbers; and merging, by the processor, the first non-compressible sequence and the second non-compressible sequence to generate the non-compressible data stream, wherein values of the non-compressible data stream alternate between values of the first non-compressible sequence and values of the second non-compressible sequence; sending, by the processor, the non-compressible data stream comprising the plurality of data blocks to be stored at a storage device, wherein the storage device is configured to store all of the plurality of data blocks in response to determining that none of the data blocks have been already stored at the storage device; receiving restored data associated with the non-compressible data stream from the storage device, wherein the restored data associated with the non-compressible data stream comprises the plurality of data blocks; determining the first constrained prime number based at least in part on a difference between a first pair of non-consecutive values from the restored data associated with the non-compressible data stream; determining the second constrained prime number based at least in part on a difference between a second pair of non-consecutive values from the restored data associated with the non-compressible data stream; and verifying an accuracy and/or reliability of the storage device, wherein to verify the accuracy and/or reliability of the storage device, the restored data associated with the non-compressible data stream is verified by using the determined first constrained prime number and the determined second constrained prime number.
 9. The method of claim 8, wherein each of the plurality of component values comprises a byte value.
 10. A computer program product, the computer program product embodied in a non-transitory computer readable storage medium and comprising computer instructions for: receiving an initialization parameter; determining a first constrained prime number and a second constrained prime number, wherein a constrained prime number comprises a plurality of component values, wherein each of the plurality of component values comprises a prime number, wherein each of the plurality of component values is different; generating a non-compressible data stream comprising a plurality of data blocks at least in part by: generating a first non-compressible sequence based at least in part on the initialization parameter and the first constrained prime number; generating a second non-compressible sequence based at least in part on the initialization parameter and the second constrained prime number, wherein the first and second non-compressible sequences are each comprised of a non-repeating sequence of numbers; and merging the first non-compressible sequence and the second non-compressible sequence to generate the non-compressible data stream comprising the plurality of data blocks, wherein values of the non-compressible data stream alternate between values of the first non-compressible sequence and values of the second non-compressible sequence; sending the non-compressible data stream comprising the plurality of data blocks to be stored at a storage device, wherein the storage device is configured to store all of the plurality of data blocks in response to determining that none of the data blocks have been already stored at the storage device; receiving restored data associated with the non-compressible data stream from the storage device, wherein the restored data associated with the non-compressible data stream comprises the plurality of data blocks; determining the first constrained prime number based at least in part on a difference between a first pair of non-consecutive values from the restored data associated with the non-compressible data stream; determining the second constrained prime number based at least in part on a difference between a second pair of non-consecutive values from the restored data associated with the non-compressible data stream; and verifying an accuracy and/or reliability of the storage device, wherein to verify the accuracy and/or reliability of the storage device, the restored data associated with the non-compressible data stream is verified by using the determined first constrained prime number and the determined second constrained prime number. 